660 research outputs found
Regular Polygonal Complexes of Higher Ranks in E^3
The paper establishes that the rank of a regular polygonal complex in 3-space
E^3 cannot exceed 4, and that the only regular polygonal complexes of rank 4 in
3-space are the eight regular 4-apeirotopes
Noncrossing partitions, clusters and the Coxeter plane
When W is a finite Coxeter group of classical type (A, B, or D), noncrossing
partitions associated to W and compatibility of almost positive roots in the
associated root system are known to be modeled by certain planar diagrams. We
show how the classical-type constructions of planar diagrams arise uniformly
from projections of small W-orbits to the Coxeter plane. When the construction
is applied beyond the classical cases, simple criteria are apparent for
noncrossing and for compatibility for W of types H_3 and I_2(m) and less simple
criteria can be found for compatibility in types E_6, F_4 and H_4. Our
construction also explains why simple combinatorial models are elusive in the
larger exceptional types.Comment: Very minor changes, as suggested by the referee. This is essentially
the final version, which will appear in Sem. Lothar. Combin. 32 pages. About
12 of the pages are taken up by 29 figure
Momentum polytopes of projective spherical varieties and related K\"ahler geometry
We apply the combinatorial theory of spherical varieties to characterize the
momentum polytopes of polarized projective spherical varieties. This enables us
to derive a classification of these varieties, without specifying the open
orbit, as well as a classification of all Fano spherical varieties. In the
setting of multiplicity free compact and connected Hamiltonian manifolds, we
obtain a necessary and sufficient condition involving momentum polytopes for
such manifolds to be K\"ahler and classify the invariant compatible complex
structures of a given K\"ahler multiplicity free compact and connected
Hamiltonian manifold.Comment: v1: 32 pages. v2: 47 pages, fixed errors, improved exposition,
expanded Section 7. v3: 47 pages, implemented changes and corrections
requested by refere
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